On group theory 3

[Papapetros Vaggelis]

Let \(\displaystyle{\left(G,\cdot\right)}\) be a group and \(\displaystyle{N}\) a normal subgroup of \(\displaystyle{\left(G,\cdot\right)}\), that is \(\displaystyle{N\trianglelefteq G}\) .

Prove that there exists a subgroup \(\displaystyle{K}\) of the group \(\displaystyle{Z(G/N)}\) such that:

\(\displaystyle{\left(Z(G)/Z(G)\cap N,\cdot\right)\simeq K\leq \left(Z(G/N),\cdot\right)}\).

[Papapetros Vaggelis]

We give a solution :

Let \(\displaystyle{f:Z(G)\longrightarrow Z(G/N)\,,f(x)=x\,N}\) .

If \(\displaystyle{x\in Z(G)}\), then \(\displaystyle{x\,y=y\,x\,,\forall\,y\in G}\) .

We will prove that \(\displaystyle{x\,N\in Z(G/N)}\). Indedd,

\(\displaystyle{x\,N\cdot y\,N=(x\,y)\,N=(y\,x)\,N=y\,N\cdot x\,N\,,\forall\,y\,N\in G/N}\) .

So, the function \(\displaystyle{f}\) is well defined.

If \(\displaystyle{x\,,y\in Z(G)}\), then

\(\displaystyle{f(x\,y)=(x\,y)N=(x\,N)\,(y\,N)=f(x)\,f(y)}\), that is, \(\displaystyle{f}\) is homomorphism.

Finally,

\(\displaystyle{\begin{aligned} \rm{Ker}(f)&=\left\{x\in Z(G): f(x)=N\right\}\\&=\left\{x\in Z(G): x\,N=N\right\}\\&=\left\{x\in Z(G): x\in N\right\}\\&=Z(G)\cap N\end{aligned}}\)

According to the 1st Isomomorphism theorem, we get :

\(\displaystyle{\left(Z(G)/Z(G)\cap N,\cdot\right)\simeq K=Im(f)\leq \left(Z(G/N),\cdot\right)}\)